Frazier Hodgson

08/12/2024 · Junior High School

An object is fired at an angle \( \theta \) to the horizontal with an initial speed of \( v_{0} \) feet per second. Ignoring air resistance, the length of the projectile's path is given by \( L(\theta)=\frac{v_{0}^{2}}{32}\left[\sin \theta-(\cos \theta)^{2} \cdot\left(\operatorname{In}\left[\tan \left(\frac{\pi-2 \theta}{4}\right)\right]\right)\right] \) where \( 0<\theta<\frac{\pi}{2} \), Complete parts (a) through (c). (a) Find the length of the object's path for angles \( \theta=\frac{\pi}{6}, \frac{\pi}{4} \), and \( \frac{\pi}{3} \) if the initial velocity is 130 feet per second. The length for \( \theta=\frac{\pi}{6} \) is \( L=481.6 \) feet. (Do not round until the final answer. Then round to the nearest tenth as needed.) The length for \( \theta=\frac{\pi}{4} \) is \( L=606.2 \) feet. (Do not round until the final answer. Then round to the nearest tenth as needed.) The length for \( \theta=\frac{\pi}{3} \) is \( L=631.2 \) feet. (Do not round until the final answer. Then round to the nearest tenth as neetged.) (b) Using a graphing utility, determine the angle required for the object to have a path length of 570 feet if the initial velocity is 130 feet per second. The angle required is \( \theta=\square \) radian(s). (Do not round until the final answer. Then round to the nearest thousandth as needed. Use a comma to separate answers as needed.)